A measure-$L^\infty$ div-curl lemma

In this note we give a very short proof of the div-curl lemma in the limit conjugate case $\mathcal M-L^\infty$, where $\mathcal{M}$ is the set of Radon measures on $\mathbb{R}^d$. The proof follows the classical approach by defining here the product in the sense of distributions via a non unique microlocal Hodge's decomposition. The result is valid for many other spaces than $\mathcal M-L^\infty$, including the classical div-curl lemma spaces $L^p-L^{p'}$ for $1<p<\infty$, and spaces of non conjugated regularity.